Denjoy example

In the theory of dynamical systems , the example of Denjoy is an example $C^{1}$ ${\ displaystyle C ^ {1}}$ $C ^ 1$ - a diffeomorphism of a circle with an irrational rotation number having a Cantor invariant set (and, accordingly, not conjugate to a pure rotation). M. Ehrman then constructed examples of such a diffeomorphism in the smoothness class $C^{1+\varepsilon }$ ${\ displaystyle C ^ {1+ \ varepsilon}}$ ${\ displaystyle C ^ {1+ \ varepsilon}}$ (i.e, $C^{1}$ ${\ displaystyle C ^ {1}}$ $C ^ 1$ with a Holder Derivative with exponent $\varepsilon$ ${\ displaystyle \ varepsilon}$ $\ varepsilon$ ) for anyone $\varepsilon <1$ ${\ displaystyle \ varepsilon <1}$ $\ varepsilon <1$ . This smoothness cannot be further increased: for diffeomorphisms with a Lipschitz derivative (and even with a derivative whose logarithm has limited variation), the Denjoy theorem holds that such a diffeomorphism with an irrational rotation number is conjugate to an irrational rotation (by the corresponding rotation number).

Content

1 Design
- 1.1 Example of homeomorphism
- 1.2 Example in the class $C^{1}$ ${\ displaystyle C ^ {1}}$ $C ^ 1$
- 1.3 Class Example $C^{1+\epsilon }$ ${\ displaystyle C ^ {1+ \ epsilon}}$ ${\ displaystyle C ^ {1+ \ epsilon}}$
2 See also
3 References
4 Literature

Design

Pasting procedure

Homeomorphism Example

The simplest example is a circle homeomorphism whose rotation number is irrational, but which, nevertheless, is not minimal . Namely, consider the rotation $R$ ${\ displaystyle R}$ at some irrational angle $\alpha$ ${\ displaystyle \ alpha}$ , and choose an arbitrary starting point $x_{0}$ ${\ displaystyle x_ {0}}$ . Consider its orbit $x_{n}=R^{n}(x_{0})$ ${\ displaystyle x_ {n} = R ^ {n} (x_ {0})}$ (for all integers $n$ ${\ displaystyle n}$ , both positive and negative). Let's make the following reorganization: at each point $x_{n}$ ${\ displaystyle x_ {n}}$ cut the circle and paste the interval $I_{n}$ ${\ displaystyle I_ {n}}$ some length $l_{n}>0$ ${\ displaystyle l_ {n}> 0}$ , so that the sum of the lengths of the glued intervals converges:

\sum _{n=-\infty }^{\infty }l_{n}<\infty .

{\ displaystyle \ sum _ {n = - \ infty} ^ {\ infty} l_ {n} <\ infty.}

Then the set obtained after such gluing will still be a circle, moreover, it will have the natural Lebesgue measure (consisting of the Lebesgue measure on the cut old circle and the Lebesgue measure on glued intervals), that is, the length - and, thus, the smooth structure. Continuing the display arbitrarily $R$ ${\ displaystyle R}$ from the old circle so that it translates the interval $I_{n}$ ${\ displaystyle I_ {n}}$ in the interval $I_{n+1}$ ${\ displaystyle I_ {n + 1}}$ , - for example, choosing as an extension the affine mapping from $I_{n}$ ${\ displaystyle I_ {n}}$ at $I_{n+1}$ ${\ displaystyle I_ {n + 1}}$ , - we get a homeomorphism f of a new circle with the same rotation number $\alpha$ ${\ displaystyle \ alpha}$ . However, this homeomorphism has a Cantor invariant set $K$ ${\ displaystyle K}$ (closure of the set of points of the old circle), and therefore it cannot be conjugated to irrational rotation.

Choosing a sequence of lengths $l_{n}$ ${\ displaystyle l_ {n}}$ so that the sequence of relationships $l_{n}/l_{n+1}$ ${\ displaystyle l_ {n} / l_ {n + 1}}$ remained limited at $n\to \pm \infty$ ${\ displaystyle n \ to \ pm \ infty}$ , for a construction with affine extension, we can achieve the Lipschitz property of the constructed homeomorphism. However, in order for the constructed map to be a diffeomorphism, the choice of extension to segments $I_{n}$ ${\ displaystyle I_ {n}}$ should be done more subtly.

Class example $C^{1}$ ${\ displaystyle C ^ {1}}$

Class example $C^{1}$ ${\ displaystyle C ^ {1}}$ is constructed so that the derivative of the constructed diffeomorphism $f$ ${\ displaystyle f}$ on the cantor set $K$ ${\ displaystyle K}$ - closure of the set of points of the original circle - would be equal to 1 (since the Lebesgue measure on this set is preserved by the constructed diffeomorphism, this is a necessary condition for such a construction). Therefore, it is necessary to choose permuting intervals $I_{n}$ ${\ displaystyle I_ {n}}$ limitations $\varphi _{n}=f|_{I_{n}}:I_{n}\to I_{n+1}$ ${\ displaystyle \ varphi _ {n} = f | _ {I_ {n}}: I_ {n} \ to I_ {n + 1}}$ so that the following conditions are met:

(D1) Derivative $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ at the ends of the interval $I_{n}$ ${\ displaystyle I_ {n}}$ equal to 1.
(D2) When $n\to \pm \infty$ ${\ displaystyle n \ to \ pm \ infty}$ derivatives of mappings $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ evenly strive for 1.

The last condition is necessary, since with growth $n$ ${\ displaystyle n}$ intervals $I_{n}$ ${\ displaystyle I_ {n}}$ accumulate to the cantor set $K$ ${\ displaystyle K}$ . Moreover, it is easy to see that these conditions are sufficient for the constructed mapping $f$ ${\ displaystyle f}$ would $C^{1}$ ${\ displaystyle C ^ {1}}$ -diffeomorphism.

By the Lagrange theorem , on the interval $I_{n}$ ${\ displaystyle I_ {n}}$ there is a point whose derivative is equal to $l_{n+1}/l_{n}$ ${\ displaystyle l_ {n + 1} / l_ {n}}$ . The second condition therefore requires that for the sequence $l_{n}$ ${\ displaystyle l_ {n}}$ took place

\lim _{n\to \pm \infty }l_{n}/l_{n+1}=1.\qquad (*)

{\ displaystyle \ lim _ {n \ to \ pm \ infty} l_ {n} / l_ {n + 1} = 1. \ qquad (*)}

As it turns out, this condition on the lengths to build $C^{1}$ ${\ displaystyle C ^ {1}}$ -diffeomorphism is also sufficient. Namely, the mappings $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ are selected as follows: on segments $I_{n}$ ${\ displaystyle I_ {n}}$ and $I_{n+1}$ ${\ displaystyle I_ {n + 1}}$ coordinates are entered that identify them with the segments $[-l_{n}/2,l_{n}/2]$ ${\ displaystyle [-l_ {n} / 2, l_ {n} / 2]}$ and $[-l_{n+1}/2,l_{n+1}/2]$ ${\ displaystyle [-l_ {n + 1} / 2, l_ {n + 1} / 2]}$ respectively, and mapping $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ selected as

\varphi _{n}=F_{l_{n+1}}^{-1}\circ F_{l_{n}},\qquad (**)

{\ displaystyle \ varphi _ {n} = F_ {l_ {n + 1}} ^ {- 1} \ circ F_ {l_ {n}}, \ qquad (**)}

Where

F_{l}:[-l/2,l/2]\to \mathbb {R} ,\quad F_{l}(x)=l\tan {\frac {\pi x}{l}}.\qquad (***)

{\ displaystyle F_ {l}: [- l / 2, l / 2] \ to \ mathbb {R}, \ quad F_ {l} (x) = l \ tan {\ frac {\ pi x} {l} }. \ qquad (***)}

A simple computation then shows that the derivative $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ at any point deviates from 1 in no more than $\mathrm {const} \cdot |1-{\frac {l_{n}}{l_{n+1}}}|$ ${\ displaystyle \ mathrm {const} \ cdot | 1 - {\ frac {l_ {n}} {l_ {n + 1}}} |}$ , therefore, condition (*) is sufficient to satisfy the second necessary condition D2. On the other hand, it is equally easy to see that condition D1 is also fulfilled (it is for this that the tangent in the formula (***) and multiplied by l: then the speed of departure to infinity at the ends $1/x$ ${\ displaystyle 1 / x}$ , and does not depend on the length of the interval l - therefore, the composition quotient concerns the identity map).

Selection of any satisfying (*) sequence $l_{n}$ ${\ displaystyle l_ {n}}$ with a convergent amount - for example, $l_{n}=1/(1+n^{2}),$ ${\ displaystyle l_ {n} = 1 / (1 + n ^ {2}),}$ - and completes the construction.

Class example $C^{1+\epsilon }$ ${\ displaystyle C ^ {1+ \ epsilon}}$

Class example $C^{1+\varepsilon }$ ${\ displaystyle C ^ {1+ \ varepsilon}}$ presented by the construction already described above, but with more subtle conditions for lengths $l_{n}$ ${\ displaystyle l_ {n}}$ . Namely, as it is easy to see, the constructed diffeomorphism will have a Hölder derivative if and only if the derivatives of all restrictions $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ uniformly across $n$ ${\ displaystyle n}$ Hölder. Indeed, comparing derivatives at points from different segments, we can subdivide this difference by derivatives at intermediate end points (since the derivative at the end point is always equal to 1) and use the triangle inequality (in the worst case, double the Hölder constant).

Since on the segment $I_{n}$ ${\ displaystyle I_ {n}}$ there is a point with a derivative $l_{n+1}/l_{n}$ ${\ displaystyle l_ {n + 1} / l_ {n}}$ (by the Lagrange theorem) there is a point whose derivative is equal to 1 (this is the end point), the Hölder constant for the Hölder exponent $\varepsilon$ ${\ displaystyle \ varepsilon}$ cannot be less than

(|1-{\frac {l_{n+1}}{l_{n}}})/l_{n}^{\varepsilon }={\frac {|l_{n+1}-l_{n}|}{l_{n}^{1+\varepsilon }}}.\qquad (L)

{\ displaystyle (| 1 - {\ frac {l_ {n + 1}} {l_ {n}}}) / l_ {n} ^ {\ varepsilon} = {\ frac {| l_ {n + 1} -l_ {n} |} {l_ {n} ^ {1+ \ varepsilon}}}. \ qquad (L)}

Therefore, the expression (L) should be limited when $n\to \pm \infty$ ${\ displaystyle n \ to \ pm \ infty}$ . It turns out that this condition of boundedness is sufficient - an explicit calculation shows that the exact Hölder constant of the constraint $\varphi _{n}$ ${\ displaystyle \ varphi _ {n}}$ differs from the lower bound (L) no more than a constant time. To complete the construction, it remains to present a two-sided infinite sequence $l_{n}$ ${\ displaystyle l_ {n}}$ with a convergent sum for which expression (L) remains bounded. An example of such a sequence is

l_{n}={\frac {1}{m\ln ^{2}m}},\quad m=2+|n|,

{\ displaystyle l_ {n} = {\ frac {1} {m \ ln ^ {2} m}}, \ quad m = 2 + | n |,}

suitable for everyone at the same time $\varepsilon <1$ ${\ displaystyle \ varepsilon <1}$ .

The presentation of such a sequence completes the construction - the constructed diffeomorphism belongs to the class $C^{1+\varepsilon }$ ${\ displaystyle C ^ {1+ \ varepsilon}}$ with any $\varepsilon <1$ ${\ displaystyle \ varepsilon <1}$ .

Links

Notes by J. Milnor Introductory Dynamics Lectures , lecture "Denjoy Theorem" (see §15B).

Literature

A.B.Katok, B. Hasselblat. Introduction to the theory of dynamical systems with a review of recent achievements / Transl. from English under the editorship of A.S. Gorodetsky. M.: МЦНМО, 2005. ISBN 5-94057-063-1
M.Herman, Sur la conjugaison différentiable des difféomorphismes du cercle à des rotations. Publications Mathématiques de l'IHÉS, 49 (1979), p. 5-233.

Denjoy example

Content

Design

Homeomorphism Example

Class exampleCone {\ displaystyle C ^ {1}}

Class exampleCone+ϵ {\ displaystyle C ^ {1+ \ epsilon}}

See also

Links

Literature

More articles:

Class example $C^{1}$ ${\ displaystyle C ^ {1}}$

Class example $C^{1+\epsilon }$ ${\ displaystyle C ^ {1+ \ epsilon}}$